Noise Reduction Using an Undecimated Discrete Wavelet Transform

         M. Lang, H. Guo,  J. E.Odegard and C. S. Burrus
        Department of Electrical and Computer Engineering
	     Rice University, Houston, TX 77251-1892

			  R. O. Wells, Jr.
	             Department of Mathematics
	       Rice University, Houston, TX 77251-1892

A new nonlinear noise reduction method is presented that uses the
discrete wavelet transform. Similar to Donoho and Johnstone, we employ
thresholding in the wavelet transform domain but, following a
suggestion by Coifman, we use an undecimated, shift-invariant,
nonorthogonal wavelet transform instead of the usual orthogonal one.
This new approach can be interpreted as a repeated application of the
original Donoho and Johnstone method for different shifts.  The main
feature of the new algorithm is a significantly improved noise
reduction compared to the original wavelet based approach, both the
$l_2$ error and visually, for a large class of signals.  This is shown
both theoretically as well as by experimental results.